Abstract:
We describe a method for obtaining estimates at infinity for eigenfunctions
of integral operators of certain classes in unbounded domains
of $\mathbb{R}^n$. We consider integral operators $K$ whose kernels
$k(x,y)$ can be written in the form $k(x,y)=a(x)k_0(x,y)b(y)$,
$(x,y)\in\Omega\times\Omega$, where
$|k_0(x,y)|\le\theta(x-y)e^{-S(x-y)}$ for some functions $\theta$ and $S$
satisfying certain natural additional conditions. We show that if the operator
$T=I-K$ with the corresponding kernel is Noetherian in $L_p(\Omega)$ and
the coefficients $a(x)$, $b(y)$ satisfy certain conditions, then the
solutions of $\varphi=K\varphi$ belong to the weighted space
$L_p(\Omega, e^{\delta S(x)})$. The method is applied to obtain
exponential estimates for eigenfunctions of $N$-particle Schrödinger
operators and estimates of decay at infinity for the solutions
of convolution-type equations with variable coefficients.
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